\newcommand{\symb}[2]{\makebox[6em][l]{#1} #2}% used to generate the list of symbols

\chapter{List of Symbols}

\symb{$i$}{Unit imaginary number; or, an index of numbers}\\
\symb{$j$, $ j' $}{Fine-structure angular momentum quantum number of individual atoms in a ground state or excited state (with prime), respectively; or, indices of numbers}\\
\symb{$f$, $ f' $}{Hyperfine-structure or hyperfine angular momentum quantum number of individual atoms in a ground state or excited state (with prime), respectively}\\
\symb{$J$, $ J' $}{The collective fine-structure angular momentum quantum number of an atomic ensemble in a ground state or excited state, respectively}\\
\symb{$F$, $ F' $}{The collective hyperfine-structure angular momentum quantum number of an atomic ensemble in a ground state or excited state, respectively}\\
\symb{$ \hat{\mathbf{S}}$, $\hat{S}_i$}{Stokes operator and its $i$-th component $ (i=0,1,2,3) $}\\
\symb{$\mathbf{e}_i$}{The unit vector along the $ i $ direction}\\
\symb{$\mathbf{u}\mathbf{v}$}{The outer product of vector $ \mathbf{u} $ and vector $ \mathbf{v} $ to form a tensor}\\
\symb{$\delta_{jk}$}{Kronecker delta function}\\
\symb{$\delta(x)$}{Dirac delta function}\\
\symb{$\dt A$}{Time derivative of $A$}\\
\symb{$A^\dagger$}{Hermitian conjugate of $A$}\\
\symb{$A^*$}{Complex conjugate of $A$}\\
\symb{$A^\transp$}{Transpose of $A$}\\
\symb{$\hat{A}^{(n)}$}{Operator $ \hat{A} $ on the $ n $th atom}\\
\symb{$\hat{\rho}^{\otimes N}$}{Tensor product of $ N $ operator $ \hat{\rho} $}\\
\symb{$\det(A)$}{Determinant of $A$}\\
\symb{$\tr(A)$, $ \tr\left[ A\right] $}{Trace of $A$}\\
%\symb{$\av{A}$}{Expectation value of $A$}\\
\symb{$[A,B]$}{Commutator of $A$ and $B$}\\
\symb{$\{A,B\}$}{Anticommutator of $A$ and $B$}\\
\symb{$\re(z)$}{Real part of $z$}\\
\symb{$\im(z)$}{Imaginary part of $z$}\\
\symb{$\mathrm{abs}[z]$, $ |z| $}{Absolute value of $z$}\\
\symb{$\identity$, $\nullmatrix$}{The identity matrix and null matrix}\\
\symb{$ \hat{\mathbbm{1}} $}{Identity operator}\\
\symb{$\unittensor$}{The identity tensor}\\
\symb{$ \expect{\hat{A}} $}{Expectation value of operator $ \hat{A} $}\\
\symb{$ \Delta A^2 $, $ \expect{\Delta A^2} $}{Variance of operator $ \hat{A} $: $ \expect{\hat{A}^2}-\expect{\hat{A}}^2 $}\\
\symb{$ \Delta A $, $ \expect{\Delta A} $}{The square root of $ \Delta A^2 $}\\
\symb{$ \expect{\Delta A \Delta B} $}{Covariance of operators $ \hat{A} $ and $ \hat{B} $}\\
\symb{$\Delta$}{Detuning}\\
\symb{$\Omega$}{Rabi frequency}\\
\symb{$\xi^2$}{metrologically relevant squeezing parameter}\\
\symb{$\chi$}{Coupling strength between atoms and light}\\
\symb{$\kappa$}{Measurement strength}\\
\symb{$\gamma_s$}{Characteristic photon scattering rate}\\
\symb{$\gamma_{op}$}{Optical pumping rate}\\
\symb{$I_{\rm in}$}{Intensity of input light}\\
\symb{$\br$,\,$\br'$}{Full spatial position vectors of a detector and the source}\\
\symb{$\br_\perp$,\,$\br'_\perp$}{The transverse spatial position vectors of a detector and the source}\\
\symb{$n_1$}{Index of refraction of the core of a waveguide}\\
\symb{$n_2$}{Index of refraction of the cladding of a waveguide}\\
\symb{$\tensor{\mathbf{G}}$}{Tensors of spatial degrees of freedom}\\
\symb{$\tensor{\mathbf{G}}(\br,\br')$}{The dyadic Green's function or Green's function tensor responded at position $ \br $ due to a source at $ \br' $}\\
\symb{$\hat{\tensor{\boldsymbol{\alpha}}}$}{Polarizability tensor operator}\\
\symb{$\Gamma_0$}{The decay rate or spontaneous emission rate of an atom in free-space}\\
\symb{$\Gamma_{\rm 1D}$}{The decay rate or spontaneous emission rate of an atom coupled to guided modes}\\
\symb{$\hat{T}^{(K)}$}{Rank-$K$ tensor operator}\\
\symb{$C^{(K)}_{j'ff'}$, $C^{(K)}_{j'f}$}{Rank-$K$ hyperfine atomic transition coefficients defined by Eqs.~\eqref{eq:Cjpffp} and~\eqref{eq:Cjpf} }\\
\symb{$C^{f_1,m_1;f_2,m_2}_{f',m'}$}{Clebsch-Gordan coefficients $ \bra{f_1,m_1;f_2,m_2}f',m'\rangle $}\\
\symb{$o^{j'f'}_{jf}$}{Relative oscillation strengths defined by Eq.~\eqref{Eq::OscStrength}}\\
\symb{$ \sigma_0 $}{On-resonance cross section of an atom}\\
\symb{$ N_A $}{Total number of atoms}\\
\symb{$ \hat{N}_C $, $ N_C $}{Number operator in the \emph{clock space} and its expectation value}\\
\symb{$ C_1 $}{Cooperativity per atom}\\
\symb{OD, OD$/ N_A $}{Optical depth, and optical depth per atom}\\
\symb{$\hat{a}$, $\hat{a}^\dagger$ }{Creation and annihilation operators for bosonic modes}\\
\symb{$ \beta $, $\beta_0$}{Propagation constant of a guided mode}\\
\symb{$ k $, $k_0$}{The total wave vector number of a propagating light}\\
\symb{$ \omega $, $\omega_0$}{Angular frequency of a light or an electromagnetic wave}\\
\symb{$\lambda$}{Wavelength of a light or an electromagnetic wave}\\
\symb{$\ket{\uparrow}$}{Fiducial state}\\
\symb{$\ket{\downarrow}$}{Coupled state}\\
\symb{$\ket{T}$}{Transfer state}\\
\symb{$\mathrm{c.c.}$}{Complex conjugate}\\
\symb{$\mathrm{H.c.}$}{Hermitian conjugate}
%\symb{$\braket{\psi}{\uppsi}$}{Annihilation operator of the state $\psi(\xbf)$, i.e., $\int\uppsi(\xbf)\,\psi^*(\xbf)\,\dif \xbf$}\\
%\symb{$\braket{\uppsi}{\psi}$}{Creation operator of the state $\psi(\xbf)$, i.e., $\int\uppsi^\dagger(\xbf)\,\psi(\xbf)\,\dif \xbf$}\\
%\symb{$\ket{\varPsi}$}{Slanted capital Greek letters for many-body quantum states}\\



